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An efficient algorithm for the least-squares reflexive solution of the matrix equation A 1 XB 1 =C 1 ,A 2 XB 2 =C 2 . (English) Zbl 1115.65048

In this paper, an iterative method for solving the minimum Frobenius norm residual problem

A 1 XB 1 A 2 XB 2 -C 1 C 2 =min

with an unknown reflexive matrix X with respect to a generalized reflection matrix P is introduced, where the matrices P and X satisfy P T =P, P 2 =I and X=XPX by definition. With any initial reflexive matrix X 1 , the matrix sequence {X k } converges to its solution within at most n 2 steps, theoretically. In addition, if

X 1 =A 1 T H 1 B 1 T +PA 1 T H 1 B 1 T P+A 2 T H 2 B 2 T +PA 2 T H 2 B 2 T P

is used for the initial reflexive matrix with arbitrary matrices H 1 ,H 2 , the solution is the least Frobenius norm solution. The numerical experiments support theoretical results.

MSC:
65F30Other matrix algorithms
15A24Matrix equations and identities